{"id":"e198955d-b757-4703-8435-4197badc72b7","arxiv_id":"2607.22470","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"For lower-triangular interacting diffusions, the N^{-1/2} fluctuation field converges to an infinite logarithmic hierarchy that couples Y^n to Y^{n+1}, not to the closed fluctuation SPDE of exchangeable mean-field systems.","lead":"Particles in this model interact only with earlier particles, and while their overall distribution converges to the same limit as a symmetric mean-field system, the size-N random fluctuations around that limit are different. The paper proves these fluctuations are described by an infinite chain of log-weighted Gaussian fields rather than by the single closed equation from classical mean-field theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The load-bearing point is the imported entropy rate (Lemma 2.1(i)) from the companion paper; if its (i-1)^{-1} decay is weaker, the replacement and remainder estimates collapse. No internal gap found; verdict unchanged.","rationale":"The paper's central claim is a joint convergence theorem for the logarithmic fluctuation hierarchy. I checked the internal structure: the tightness argument (Proposition 3.3), the replacement of T_N h_n by h_{n+1} (Lemma 4.3), the covariance computations (Lemma 4.5, Proposition 4.6), the vanishing of the nonlinear remainder (Proposition 5.2), and the weighted Volterra uniqueness (Proposition 6.1) are all coherent, provided Lemma 2.1 holds. No circularity or fitted-versus-predicted reasoning appears. The weakest point is indeed the external entropy estimate imported from [35]: it is used at several key steps, and the paper does not reproduce its proof. The reader's verdict already flags this. However, the reader slightly overstates the import: Lemma 2.1(iv) is proved in this manuscript, while the truly unproved, load-bearing item is Lemma 2.1(i). Since the paper explicitly cites the companion and the rest of the argument is self-contained given that estimate, I do not see a reason to change the ACCEPT verdict. The recommended concrete verification is to re-derive the incremental entropy rate from [35, Lemma 3.1] and, if necessary, test the consequences of a slower rate in the estimates of Proposition 5.2.","tokens_in":27589,"tokens_out":34664,"duration_ms":349612,"concrete_test":"Inspect the companion paper [35]; take its Girsanov identity (Lemma 3.1) with the i.i.d. initial law and recompute the incremental entropy R_i(T)=H(P^{1:i}_{[0,T]} | P^{1:i-1}_{[0,T]} ⊗ Pbar_{[0,T]}). Verify the bound R_i(T) <= C_T/(i-1) with C_T independent of i and N. If the calculation yields a slower rate, re-run Proposition 5.2 term D and Lemma 4.3 bias with that rate: any TV decay (i-1)^{-alpha} with alpha < 1/4 will make the D-term diverge and invalidate the proof of Theorem 1.3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 2.1(i) is the true load-bearing imported input: R_i(T) <= C_T/(i-1) and E∫|Delta_i|^2 <= C_T/(i-1). These feed Lemma 3.2, Proposition 3.3, Lemma 4.3 (via the TV bound), Lemma 5.1, and term D in Proposition 5.2. If the decay were only (i-1)^{-alpha} with alpha < 1/2, the bias term in Lemma 4.3 would scale like N^{-1/2} Σ |e_n(i/N)| (i-1)^{-alpha}; while for alpha=1/2 this vanishes, for alpha < 1/4 the term D in Proposition 5.2 can diverge. The manuscript does not prove (i); it cites [35]. Lemma 2.1(iv) is actually proved in Section 2.2, so the unproved core is smaller than the reader's wording suggests, but the entropy rate remains unverified here. This is a disclosed dependency, not an internal inconsistency.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the lower-triangular (sequential) interacting particle system (1.1), in which particle i interacts only with its predecessors. It introduces a countable family of logarithmically weighted empirical fluctuation fields Y^{N,n}, n≥0, and proves (Theorem 1.3) that, under a smooth convolutional interaction condition, the entire family converges jointly in the countable product of weighted negative Sobolev path spaces to the unique probabilistically strong solution of a linear hierarchy in which level n couples to level n+1. The limiting interaction is not the classical closed McKean–Vlasov fluctuation equation but rather a logarithmic hierarchy; the paper also identifies the limiting Gaussian initial law and martingale covariances explicitly. The proof combines exact finite-N identities, entropy and conditional-law estimates from the companion paper [35], deterministic Riemann-sum asymptotics for the logarithmic weights, tightness, conditional replacement for the nonlinear remainder, and a weighted Volterra Gronwall argument for uniqueness.","tokens_in":27878,"tokens_out":35268,"duration_ms":396446,"significance":"If the result holds, it is a substantial contribution to the fluctuation theory of non-exchangeable interacting diffusions. The contrast with the column-balanced universal CLT of Shkolnikov–Yeung is clearly demonstrated and is conceptually interesting: the law-of-large-numbers limit is the same as in the exchangeable case, but the N^{-1/2} fluctuations reveal the sequential structure through the nonclosed hierarchy of logarithmic weights. The paper contains several strong, explicitly checkable ingredients: exact finite-N identities (Lemmas 4.1 and 4.2), explicit limiting covariances via logarithmic Riemann sums (Lemma 2.6 and Proposition 4.6), a conditional-replacement estimate that removes the nonlinear remainder (Proposition 5.2), and a pathwise uniqueness proof for the limiting hierarchy (Proposition 6.1). The exposition is transparent about the main external input, Lemma 2.1(i), which is imported from the same authors' companion paper [35].","major_comments":[{"comment":"The two estimates R_i(T) ≤ C_T/(i−1) and E∫_0^T |Δ_i|^2 dt ≤ C_T/(i−1) are load-bearing: they enter the tightness bound (Proposition 3.3), the replacement of T_N h_n by h_{n+1} (Lemma 4.3), the martingale covariance identification (Lemma 4.5), and the vanishing of all four terms in the nonlinear remainder (Proposition 5.2). The proof of Lemma 2.1(i) here is only a citation to [35, Theorem 1 and Lemma 3.1], and the statement of the exact theorem from [35] is not reproduced. This is a disclosed, non-circular dependency, but it is nevertheless the quantitative foundation of the paper. I recommend that the authors either state precisely which result in [35] yields the decay and include its main assumptions, or reproduce a proof sketch in an appendix; otherwise the referee and readers cannot independently verify the core convergence argument.","section":"Section 2.2, Lemma 2.1(i)"},{"comment":"The proof of Lemma 4.4 relies on the assertion that K ∈ C_b^∞ ∩ L^1 implies K ∈ H^r for every r ≥ 0, via Gagliardo–Nirenberg interpolation. This is plausible but is stated in one sentence and is important: the bound (4.9) is used to show that the interaction kernel Γ_s^φ is in H^{β*} and to control the cutoff error with (1−χ_R)Γ_s^φ. Please provide the precise interpolation inequality being applied, including the role of the uniform bounds on all derivatives of K, so that the reader can verify that the needed H^r regularity does not require additional decay assumptions on K.","section":"Section 4.2, Eq. (4.8)–(4.9)"}],"minor_comments":[{"comment":"The text contains several OCR-like typesetting artifacts (e.g., 'N −1/2', 'D M N,n', 'M N n', 'R i(s)'). The final version should be carefully typeset.","section":"General typesetting"},{"comment":"In the part j > εN, the text says that (1/√N) Σ_{j>εN} j^{−1/2} → 0; in fact this factor is bounded by 2(1−√ε)+o(1), not o(1). The conclusion still follows because sup_{j>εN}|e_N(j)| → 0, but the wording should be clarified.","section":"Lemma 2.7, proof of the predecessor-weighted L1 bound"},{"comment":"The condition q ∈ (0,1/4) is used to absorb the 4^n energy bound from Proposition 3.3. It would be helpful to say explicitly that any q < 1/4 is admissible, since Proposition 6.1 again chooses such a q.","section":"Definition 1.2, Eq. (1.4)"},{"comment":"References [29] and [35] are 2026 arXiv preprints. Please update them to published versions if available, and confirm that [35] is publicly accessible and contains the exact theorem cited in Lemma 2.1(i).","section":"References"},{"comment":"There is a typo in the affiliation block: 'Zhenfu W ang' should be 'Zhenfu Wang'.","section":"Author line"}],"recommendation":"major_revision","confidential_remarks":"The mathematical core is original and the finite-N algebra is elegant. I do not find an internal inconsistency, and the stress-test concern about Lemma 2.1(i) is correctly described as a disclosed dependency. However, that lemma is so central to tightness, replacement, and remainder estimates that the manuscript should either reproduce the proof or give a precise, verifiable reference to the companion paper. The Gagliardo–Nirenberg step in Lemma 4.4 also needs a small expansion. These are fixable within the scope of the paper, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper proves something genuinely new: for the lower-triangular interacting diffusion system, the N^{-1/2} fluctuation limit is not the classical McKean–Vlasov fluctuation SPDE but the first level of an infinite hierarchy of logarithmically weighted fields. The hierarchy never closes: level n couples to level n+1, and the weights are exactly h_n(u) = (log(1/u))^n/n!. This is a real discovery, and it sharpens the boundary of Shkolnikov–Yeung's universal CLT, which excludes the sequential column sums because they diverge. I believe the paper delivers on this claim.\n\nWhat is good: the finite-N algebra is explicit and exact (Lemmas 4.1, 4.2, and the predecessor-sum identity). The limiting covariances for initial fields and martingale noises are computed via clean Riemann sums. The tightness and identification argument is coherent; the conditional replacement in Lemma 5.1 really uses the conditional-law kernels, not hand-waving. And they prove strong well-posedness of the limiting hierarchy with a Volterra Gronwall estimate. That's a complete package.\n\nThe soft spot is exactly what the reader flagged: Lemma 2.1(i), the incremental entropy bound R_i(T) <= C/(i-1) with the integrated drift error, is imported from the companion paper [35] without proof. That bound is load-bearing—it enters the tightness estimate (Prop 3.3), the replacement (Lemma 4.3), and the vanishing of the nonlinear remainder (Prop 5.2). The stress-test note is right that if the decay were weaker than (i-1)^{-1/2}, the argument would collapse. But this is a disclosed and standard-looking dependency, and the paper proves Lemma 2.1(iv) itself. I do not see the importation as a gap; it is a clear pointer to a companion result that presumably contains the proof. A referee would need to check it, which is normal.\n\nMy own take is slightly more positive than the reader's ACCEPT with moderate confidence: the central claim is not circular, and the hierarchy is explicit enough to be independently checked. The comparison with Shkolnikov–Yeung is honest and precise. The only real risk is the companion paper's entropy rate, and the authors have flagged it themselves.\n\nWho is this for? Anyone working in mean-field fluctuation theory, non-exchangeable propagation of chaos, or stochastic interacting systems. It deserves a serious referee. I would send it to review, and I would cite it if I were working in the area.","headline":"A genuinely new fluctuation phenomenon—an infinite logarithmic hierarchy—proved cleanly, with the only real vulnerability being an entropy estimate imported from the authors' companion paper.","tokens_in":28355,"tokens_out":2469,"would_cite":true,"duration_ms":25979,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","60K35","60H15","60J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sequential order survives at the fluctuation scale: an infinite hierarchy, not a closed equation.","keywords":["sequential interacting diffusions","non-exchangeable particle systems","Gaussian fluctuations","central limit theorem","logarithmic fluctuation hierarchy","mean-field limit","fluctuation SPDE"],"falsifier":"Simulate the lower-triangular system in one dimension with a smooth kernel and estimate the finite-N cross-covariance of the unweighted field Y^{N,0} and the log-weighted field Y^{N,1} at a fixed time. The hierarchy predicts a nonzero limiting coupling between the two levels, whereas the classical closed equation predicts that Y^{N,1} is asymptotically irrelevant to the drift of Y^{N,0}; large-N agreement with the paper's covariance supports the hierarchy, and agreement with the closed equation would refute it.","tokens_in":27468,"feed_emoji":"🪜","tokens_out":6772,"duration_ms":80824,"temperature":0.7,"pith_summary":"Sequential interactivity changes the central-limit corrections of a mean-field diffusion even when it does not change the law of large numbers. In this system, particle i feels only particles 1 through i-1, yet the empirical measure converges to the same deterministic limit as a fully symmetric mean-field system. The paper proves that at the 1/sqrt(N) scale the ordering remains visible: the averaged fluctuation field couples not to itself but to a field weighted by log(N/i), which couples to the next logarithmic weight, and so on. The whole countable family converges jointly to a unique linear hierarchy, with explicit covariances. If correct, this shows that fluctuations encode interaction geometry that the deterministic limit washes out.","feed_headline":"Sequential particle fluctuations form an infinite hierarchy","feed_subtitle":"Each fluctuation level feeds the next, exposing order that the law of large numbers hides.","key_machinery":"The central object is the logarithmic fluctuation ladder Y^{N,n} = Y^N(h_n) with h_n(u) = (log(1/u))^n / n!, together with the discrete predecessor-sum operator (T_N f)(j/N) = sum_{i=j+1}^N f(i/N)/(i-1). Reversing the order of the lower-triangular interaction sum rewrites the drift of level n as a column-sum weight (T_N h_n)(j/N) on particle j, and the deterministic lemma T_N h_n -> h_{n+1} shows the level-n equation depends on level n+1. The ladder never closes; the proof controls the unbounded weights near u=0 by weighted Sobolev norms and q^n-summed energy bounds.","core_discovery":"Under smooth drift, a convolution kernel, and i.i.d. initial conditions, the paper establishes Theorem 1.3: the family Y^{N,n}_t = N^{-1/2} sum_i ((log(N/i))^n / n!) (delta_{X^i_t} - rho_bar_t), n >= 0, converges jointly in a countable product of weighted path spaces to the unique probabilistically strong solution of a linear hierarchy in which each level n satisfies d<Y^n,phi> = <Y^n,L phi> dt + <rho_bar_t,(K*Y^{n+1}).grad phi> dt + dM^n(phi). The coupling is the predecessor-summation limit: reversing the order of summation turns the interaction with earlier particles into a cumulative weight that converges to log(1/u), and iterating produces (log(1/u))^n / n!, so no finite block of equatio","pith_inferences":["By the same column-sum mechanism, other non-exchangeable graphs should produce hierarchies generated by their own cumulative-weight profile; the logarithmic ladder is the special case of uniform sequential order, so the method suggests a way to read interaction geometry from fluctuation covariances.","The combinatorial factor ((n+k)!/n!k!) equals moments of a standard exponential variable, hinting at a limiting log-time representation; one could test whether the joint law of the hierarchy matches a Gaussian field built from iterated integrals of a single noise.","A practical corollary is that early particles are informationally dominant at the fluctuation scale: their cumulative column weights diverge, so statistical estimators for the mean-field limit should down-weight early labels less than exchangeability would suggest.","Because the level-0 drift coefficient is literally the column-sum profile, one could in principle estimate the interaction ordering from time-series data of fluctuations; a testable prediction is that the cross-covariance between Y^0 and Y^1 vanishes only under column balance."],"forward_implications":["If the theorem is right, the classical closed fluctuation SPDE is not the universal second-order description of mean-field diffusion: two systems with the same deterministic limit can have different Gaussian fluctuations.","Fluctuation limits distinguish interaction architectures: balanced weights give constant column sums and the classical limit, while uniform sequential weights give a logarithmic profile, with an explicit L2 gap separating the regimes.","Every finite collection of levels is non-closed; any approximation of the fluctuation field must either carry the whole ladder or introduce an error controlled by the q^n weights.","Strong well-posedness plus the uniqueness argument means the full sequence converges, not just subsequences, and the limiting Gaussian field is a well-defined statistical object.","The explicit covariance formula ((n+k)!/n!k!) gives a directly testable signature of the hierarchy."],"fun_headline_variants":["Sequential fluctuations form an infinite chain","Order at fluctuation scale: a hierarchy","Each fluctuation level feeds the next","Predecessor order unlocks fluctuation hierarchy","Infinite fluctuation cascade from sequential diffusions"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole proof depends on imported quantitative estimates that each particle's conditional law is within about 1/sqrt(i) of the mean-field law in total variation and that centered interaction errors have a sub-Gaussian tail; if either rate fails, the tightness and nonlinear-remainder steps collapse.","fun_headline_variants_meta":{"raw":{"variants":["Sequential fluctuations form an infinite chain","Order at fluctuation scale: a hierarchy","Each fluctuation level feeds the next","Predecessor order unlocks fluctuation hierarchy","Infinite fluctuation cascade from sequential diffusions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000276,"raw_usage":{"total_tokens":1513,"prompt_tokens":803,"completion_tokens":710,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":547,"completion_tokens_details":{"reasoning_tokens":649}},"tokens_in":547,"tokens_out":710,"duration_ms":7669,"temperature":1.0,"reasoning_tokens":649,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T04:36:49.023351+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the lower-triangular system in one dimension with a smooth kernel and estimate the finite-N cross-covariance of the unweighted field Y^{N,0} and the log-weighted field Y^{N,1} at a fixed time. The hierarchy predicts a nonzero limiting coupling between the two levels, whereas the classical closed equation predicts that Y^{N,1} is asymptotically irrelevant to the drift of Y^{N,0}; large-N agreement with the paper's covariance supports the hierarchy, and agreement with the closed equation would refute it.","supporting_citations":[],"review_version":1}